Some remarks on the local class field theory of Serre and Hazewinkel. (English. French summary) Zbl 1362.11096
Summary: We give a new approach for the local class field theory of J.-P. Serre [Bull. Soc. Math. Fr. 89, 105–154 (1961; Zbl 0166.31103)] and Hazewinkel [M. Demazure and P. Gabriel, Algebraic groups. Volume I: Algebraic geometry. Generalities. Commutative groups. With an appendix ‘Local class fields’ by Michiel Hazewinkel. Paris: Masson et Cie (1970; Zbl 0203.23401)]. We also discuss two-dimensional local class field theory in this framework.
MSC:
| 11S31 | Class field theory; \(p\)-adic formal groups |
| 11R37 | Class field theory |
| 11S70 | \(K\)-theory of local fields |
References:
| [1] | 2 [[S,T]] is compatible with the group structure if µ * M ∼ = pr * 1 M ⊗ pr * 2 M , where µ : Ω 2 [[S,T]] × Ω 2 [[S,T]] → Ω 2 [[S,T]] is the addition map and pr i is the i-th projection for each i = 1, 2. Let C be the category of D-modules of O-rank one on Ω 2 [[S,T]] that are compatible with the group structure and let C be the category of D-modules of O-rank one on Spec K. The categories C and C are k-linear Picard categories under the tensor operation. The endomorphism ring of any object of C or C is equal to k. Let Isom( C) (resp. Isom( C )) be the abelian group of isomorphism classes of objects of C (resp. C ). Since Ω 2 [[S,T]] |
| [2] | and Spec K are the Spec’s of unique factorization domains, line bundles on them can be trivialized. We can associate connection forms to objects of C and C whose underlying line bundles are trivial. Thus Isom( C) (resp. Isom( C )) can identified with a subquotient (as an abelian group) of the space of 1-forms on the k-scheme Ω 2 [[S,T]] (resp. Spec K). |
| [3] | The pullback by ϕ 1 : Spec K → Ω 2 [[S,T]] gives a fully faithful embedding C → C of k-linear Picard categories. |
| [4] | Under the above identification of C , we have Isom( C ) ∼ = (k((T ))/Z) dlog S ⊕ (k((S))/Z) dlog T |
| [5] | ⊕ T −1 Sk[[S]][T −1 ] ⊕ T −1 S −1 k[S −1 , T −1 ] . |
| [6] | The image of Isom( C) in Isom( C ) by the pullback functor by ϕ 1 corre-sponds to the last summand d(T −1 S −1 k[S −1 , T −1 ]) via the isomorphism of Assertion 2. |
| [7] | Proof. -2. The group Isom( C ) is identified with Ω 1,d=0 · Zbl 0931.41017 |
| [8] | K/k / dlog K × . A straight- |
| [9] | K/k / dlog K × is equal to the right-hand side of Assertion 2. |
| [10] | The above proof of Assertion 3 also shows that the homomorphism |
| [11] | Isom( C) → Isom( C ) induced by the functor C → C is injective. The result follows from this. BIBLIOGRAPHY |
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